| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Product with reciprocal term binomial |
| Difficulty | Moderate -0.8 This is a straightforward binomial expansion question requiring routine application of the binomial theorem with a reciprocal term. Part (i) involves direct calculation of three terms using the formula, and part (ii) requires identifying which terms from part (i) contribute to x^4 after multiplication—a standard technique. The question is easier than average as it involves no problem-solving insight, just methodical application of a well-practiced procedure. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\left(x - \frac{2}{x}\right)^6 = x^6 - 12x^4 + 60x^2\) | B1 ×3 | co |
| [3] | ||
| (ii) \(\times (1 + x^2) \to 60 - 12 = 48\) | M1 A1√ | Must be exactly 2 terms. √ from his (i). |
| [2] |
(i) $\left(x - \frac{2}{x}\right)^6 = x^6 - 12x^4 + 60x^2$ | B1 ×3 | co
| [3] |
(ii) $\times (1 + x^2) \to 60 - 12 = 48$ | M1 A1√ | Must be exactly 2 terms. √ from his (i).
| [2] |
---2 (i) Find the first three terms, in descending powers of $x$, in the expansion of $\left( x - \frac { 2 } { x } \right) ^ { 6 }$.\\
(ii) Find the coefficient of $x ^ { 4 }$ in the expansion of $\left( 1 + x ^ { 2 } \right) \left( x - \frac { 2 } { x } \right) ^ { 6 }$.
\hfill \mbox{\textit{CAIE P1 2010 Q2 [5]}}